MPA Cosmology Lecture Series: Modified Gravity Alex Barreira 1 Goal of this lecture . These lecture notes aim to provide with a broad introduction to the study of modified gravity models in a cosmological context. The main objective is to draw a picture of the research field as a whole, ranging from theoretical to more phenomenological/observational aspects. Owing to the often lengthy calculations, sometimes we will be forced to restrict ourselves to displaying only the final result and interpret the solutions qualitatively. Whenever possible though, we shall go through some calculations with a bit more detail, both for fun and intuition-building purposes. The diagram of Fig. 1 displays the outline (which is also a summary) of this lecture. For these notes, although it is desirable that one is familiar with variational calculus and tensor algebra, this is not mandatory. It is nearly impossible to avoid discussing modified gravity without resorting to actions and Lagrangians, but care was taken such that whenever and action is written, it is immediately followed by the associated equations of motion (which may perhaps be easier to interpret for some). It is very unlikely that someone will become a field expert by just reading these notes. Further reading suggestions include the following reviews (on which parts of these notes are based): • Clifton, Ferreira, Padilla & Skordis, arXiv:1106.2476. Being over 300 pages long and referring to over 1300 papers, this is by far the most comprehensive review of modified gravity in cosmology. Most of the focus is on the theoretical aspects of the models, and less so on their observational signatures. • Joyce, Jain, Khoury & Trodden, arXiv:1407.0059. This review organizes the discussion by the dif- ferent types of screening mechanisms, discussing their theoretical aspects as well as typical observational tests and constraints. • Koyama, arXiv:1504.04623. The structure and scope of this review is similar to the one above, but it pays more attention to the observational aspects of modified gravity (in particular at the nonlinear level of structure formation). 2 Motivation for modified gravity studies In the current standard ΛCDM cosmological model, the gravitational interaction is described by Einstein's theory of General Relativity (GR). The main reason for this is perhaps related to its remarkable agreement with a wealth of precision tests of gravity done in the Solar System. These include the classical tests of gravitational redshift, the lensing of the light from background stars by the Sun and the anomalous perihelion of Mercury, as well as other tests such as the Shapiro time-delay effect measured by the Cassini spacecraft and Lunar laser ranging experiments which meausure the rate of change of the gravitational strength in the SS. Outside of the SS, GR is also in good agreement with the tests that involve changes in the orbital period of binary pulsars due to the emission of gravitational waves. Despite of these tremendous successes, however, one can still think of a few reasons to expect/suspect/wish that GR does not provide us with the full picture: 1 Figure 1: Outline/summary of the content of these notes. • Is GR correct on large scales? The abovementioned tests probe the gravitational law only on scales smaller than the Solar System. This means that the application of GR in any cosmological study consti- tutes in fact a huge extrapolation of the regime of validity of the theory. In other words, there is room for deviations from GR on cosmological scales, and the size of such deviations should be constrained. • Modified gravity can be dark energy. Dark energy is the general name given to any form of energy with negative enough pressure to have "repulsive" gravity. Its existence is postulated to explain the observed accelerated expansion of the Universe, which is otherwise impossible in a Universe governed by GR and containing only the matter species we know (radiation and matter). The argument for modified gravity is that the need to postulate dark energy may follow from our wrong use of GR as the theory of gravity on large scales. In other words, what we think are effects of dark energy may simply be the effects of the corrections to GR that we are still unaware of. • GR has no quantum limit. A final (slightly more speculative, but enlightening) way to gain courage to go ahead and modify GR is to remind ourselves that this theory does not have a well defined quantum field limit. Taking for granted that all interactions must have a quantum field description, then GR cannot be the final answer and must be corrected. To be fair, these corrections to GR must take place on small scales or in the high-energy limit, whereas in cosmology we are concerned with the opposite end of the energy spectrum: weak fields on large scales. Nevertheless, it is not unreasonable to believe that an eventual quantum field theory of gravity that differs from GR on small scales, should also differ from it on cosmological ones. 3 What is modified gravity? In this section, we shall try to specify what modified gravity actually means. First of all (and as we could guess from the discussion in the previous section), the phrase "modified gravity" is a slight abuse 2 of language. It is used to describe any theory of gravity that goes beyond GR, and so "modified GR" would be a more appropriate name. As a result, the best way to start defining modified gravity is with a quick recap of GR. 3.1 General Relativity in a nutshell GR can be described by the action Z p R S = d4x −g − L ( ; g ) ; (1) 16πG m µν where g is the determinant of the 2-rank metric tensor field gµν, R is called the Ricci scalar, Lm is the Lagrangian density that describes the forms of energy we know (dark matter, baryons, radiation, etc., described collectively by the field ) and the integration is taken over the whole four-dimensional µ spacetime x (µ = 0; 1; 2; 3). By varying this action w.r.t. gµν (in these notes, we will not be bothered with the boring algebra of variational calculus, and will just take the result for granted), we arrive at the famous Einstein field equations 1 G ≡ R − Rg = 8πGT m ; (2) µν µν 2 µν µν where Rµν is the Ricci tensor and Tµν the energy-momentum tensor associated with Lm. The left hand- side of this equation contains purely geometric terms (i.e., the metric and its derivatives), whereas the right hand-side specifies the energy content that exists in the Universe. For the reader that is least familiar with tensor algebra, the quick way to interpret Eq. (2) is to think of it as a set of 16 equations, each labelled by (µ, ν). In fact, all the above tensors are symmetric, i.e., Tµν = Tνµ, which means that there can only be 10 different equations (some of these 10 equations can also be redundant, depending on the exact application in mind). Another important aspect of GR is that the Einstein tensor Gµν is divergence-free, which naturally ensures energy-momentum conservation: µν µν rνG = 0 =) rνT = 0: (3) In order to arrive at a concrete set of equations to work with, one needs to specify two things: (i) the energy-momentum tensor { to plug in the right-hand side of Eq. (2); and (ii) the metric { to define the curvature tensors on the left-hand side. In cosmology, it is common to take the form of Tµν to be that of a pefect fluid Tµν = (ρ + P ) uµuν − P gµν; (4) where ρ, P and uµ are, respectively, the density, the pressure and the four-velocity of the fluid. For completeness, we note that we are neglecting the fluid’s heat flux (a vector) and anisotropic stress (a tensor) in the above equation, but this is not critical for these notes. Motivated by the cosmological principle, the line element of the metric field is taken to be that of a Friedmann-Robertson-Walker (FRW) spacetime ds2 = (1 + 2Ψ) dt2 − a(t)2 (1 − 2Φ) dx2 + dy2 + dz2 ; (5) where Ψ, Φ are two gravitational potentials and a = 1=(1+z) is the scale factor (z is the redshift). At the background level, Ψ = Φ = 0, and a becomes the only variable to solve for (the "size" of the spatial sector of the Universe as it expands). Before proceeding, it is important to mention that we are considering only scalar perturbations (Ψ; Φ are scalar fields) to the homogeneous FRW picture. In general, Eq. (5) can contain also vector (which are typically very small and decay rather quickly) and tensor perturbations (gravitational waves). Vector and tensor perturbations are not covered in these notes. We have also assumed that the Universe is spatially flat. 3 3.2 Key equations At the background level (recall Φ = Ψ = 0 and uµ = (−1; 0; 0; 0)), using the (0; 0) component and (i; i) µν component (i = 1; 2 or 3) of Eq. (2), as well as the 0-component of rµT = 0, we arrive at a_ 2 H2 ≡ = 8πGρ,¯ (6) a a¨ 4πG = − ρ¯ + 3P¯ ; (7) a 3 ρ¯_ = −3H ρ¯ + P¯ ; (8) where an overdot denotes a derivative w.r.t. physical time t and an overbar indicates background quan- tities. Given the matter content, these equations specify the rate at which the Universe expands. For −3 instance, in a matter dominated universe (¯ρ =ρ ¯m, P¯ = 0), Eq. (8) tells us thatρ ¯m =ρ ¯m0a , where we defineρ ¯m0 as the present-day value (a = 1) of the matter density.